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Properties

Label 2-3024-28.23-c0-0-2
Degree $2$
Conductor $3024$
Sign $0.997 - 0.0633i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 13-s + (0.5 + 0.866i)25-s + (−1.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 1.73i·43-s + 49-s + (0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + (1 + 1.73i)73-s + (1.5 + 0.866i)79-s + 91-s + 97-s + (−1.5 − 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯
L(s)  = 1  + 7-s + 13-s + (0.5 + 0.866i)25-s + (−1.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 1.73i·43-s + 49-s + (0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + (1 + 1.73i)73-s + (1.5 + 0.866i)79-s + 91-s + 97-s + (−1.5 − 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :0),\ 0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.467451750\)
\(L(\frac12)\) \(\approx\) \(1.467451750\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.820081740306154960590833537886, −8.263826248559455055939129373699, −7.39675079304989587397255431417, −6.79437257771431562415112274757, −5.64876366594014347599404886442, −5.23189641859203798204520284516, −4.14573848771854729491142315653, −3.45458522418613692084503487514, −2.18042506411695364806450510456, −1.24043189277363175749018302102, 1.18556202610363121711901065780, 2.20385379468149205852415791300, 3.34431945495599445606938385071, 4.28831107292738707892838601364, 4.96445288068687993402731273053, 5.91473502389928183082332716287, 6.53093312987643819916560418355, 7.63300297453339727998686477078, 8.059029675906616316052115662124, 8.866525362048845582821812445013

Graph of the $Z$-function along the critical line